Problem: You have found the following ages (in years) of 4 seals. The seals are randomly selected from the 20 seals at your local zoo: $ 32,\enspace 2,\enspace 4,\enspace 12$ Based on your sample, what is the average age of the seals? What is the variance? You may round your answers to the nearest tenth.
Because we only have data for a small sample of the 20 seals, we are only able to estimate the population mean and variance by finding the sample mean $({\overline{x}})$ and sample variance $({s^2})$ To find the sample mean , add up the values of all $4$ samples and divide by $4$ $ {\overline{x}} = \dfrac{\sum\limits_{i=1}^{{n}} x_i}{{n}} = \dfrac{\sum\limits_{i=1}^{{4}} x_i}{{4$ To compensate for this underestimation, rather than simply averaging the squared deviations from the mean , we total them and divide by $n - 1$ $ {s^2} = \dfrac{\sum\limits_{i=1}^{{n}} (x_i - {\overline{x}})^2}{{n - 1}} $ $ {s^2} = \dfrac{{380.25} + {110.25} + {72.25} + {0.25}} {{4 - 1}} $ $ {s^2} = \dfrac{{563}}{{3}} = {187.67\text{ years}^2} $ We can estimate that the average seal at the zoo is 12.5 years old. There is a variance of 187.67 years $^2$.